Journal on Vehicle Routing Algorithms (2018) 1:105–119https://doi.org/10.1007/s41604-018-0008-3

REGULAR PAPER

Heuristics for the weighted k-rural postman problemwith applicationsto urban snow removal

Kaj Holmberg1

Received: 3 April 2017 / Accepted: 7 February 2018 / Published online: 7 March 2018© The Author(s) 2018. This article is an open access publication

AbstractWe describe a weighted version of the k-Chinese and k-rural postman problem that occurs in the context of snow removal. Theproblem concerns the questions of which vehicle shall take care of each link and how the vehicles shall travel between links.We also consider different numbers of vehicles, in view of a fixed cost for each vehicle. We describe and discuss heuristicsolution approaches, based on usable substructures, such as heuristics for rural postman problems, meta-heuristics, k-meansclustering and local search improvements by moving cycles. The methods have been implemented and tested on real lifeexamples.

Keywords k-Chinese postman problem · Rural postman problem · Heuristics · Clustering

1 Introduction

Snow removal can be an important and costly issue, in cer-tain places at certain times. The yearly amounts of snow hasrecently varied much in several countries. Some years thereis a lot, other years hardly anything. Hence, the demand ofsnow removal varies a lot. Snow removal can be quite dif-ficult and expensive, if there is very much snow. Winterswithout snow make it difficult to motivate keeping a largefleet of snow removal vehicles available. Complaints fromthe public about snow removal are common. A conclusion isthat one cannot expect to use the plans from last year. Newsituations require new plans, which require planning tools.

Let us describe the planning situation (in Sweden). A cityis divided into areas, and for each area, a contractor is hired.As the contractors work independently, such an area givesnatural borders for the optimization. Each contractor has anumber of vehicles/drivers that togetherwill remove the snowfrom all streets in the area. Each driver is allocated a set ofstreets, and in practice is often free to plan the tour by himself.Good tours are often based on years of experience, but thismay fail when new drivers are hired, or if there has been verylittle snow for many years. Another aspect that might not be

B Kaj Holmbergkaj.holmberg@liu.se

1 Department of Mathematics, Linköping Institute ofTechnology, SE-581 83 Linköping, Sweden

optimal is that the drivers’ designated areas are for practicalreasons usually as separate as possible, since their tours arenot coordinated.

The treatment of a street includes many details, such asseveral sweeps, depending on the width of the street, addi-tional clearing at crossings, and even considerations of thefact that it might take a longer time turning a vehicle aroundthan continuing straight ahead. Such details are taken intoaccount in a huge MIP-model presented in the technicalreport [15]. Unfortunately, that MIP-model is practicallyimpossible to solve. In [15] we also present a number ofrelaxations of the model, which may be used to get boundson the optimal objective function, but do not give feasiblesolutions to the problem.

Let us now describe the problem to be solved here. Weconsider the area assigned to one contractor, and focus onthe question how to allocate the streets to the vehicles. Toevaluate an allocation, we need to find tours for the vehicles.However, there may in practice always be small disturbances(parked cars, more or less snow than expected, etc.) that leadto deviations from the planned tours. Therefore, we see theallocation of streets to vehicles as the most important part ofa solution.

We will do the optimization in stages, first find an allo-cation of streets to vehicles, and then find good tours forthe vehicles. Obviously, we need to iterate and modify theallocation based on the costs of the tours.

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When a good allocation is finally at hand, a tour taking alldetails into accountwill be produced.That step is described in[19]. Therefore, we here omit many details, and approximatethe work needed to clear a street by one operation, since thepurpose of the tour is mainly to evaluate the allocation.

We assume that the vehicles are identical, and do not usedepots, since we do not know where the vehicles will start.In practice, this means that a vehicle can enter the tour at anynode.

Finding data for the problem can be an issue. The streetnetwork and all link characteristics can nowadays be foundin different databases. In this paper, we use data from Open-StreetMap. In [18], we give more details about what datathere is and how to extract the data to a useful form.

Times for doing tasks are known approximately, based onan average speed. The operators will in the future be requiredto equip each vehicle with a GPS. This gives data fromwhichwe may extract times for all tasks. Therefore, task times willbe available with better accuracy. Analyzing the GPS tracksto extract that information will require map matching, whichis described in [17].

We can solve the problem for different numbers of vehi-cles, as a way to dimension the fleet of vehicles. Clearly, itis faster to let several vehicles work in parallel, but there isa fixed cost for each vehicle that does not depend on howmuch the vehicle is used, only on the fact that it is available.

What kind of optimization problem is this? If there wasonly one vehicle to do all snow removal, the problem wouldbe reduced to a Chinese postman problem, which simply isto find a shortest round trip that covers all links in a graph.

Related problems are the following. The rural postmanproblem is to find a shortest round trip that covers certainlinks. This corresponds to one vehicle clearing a subset ofthe links. The k-Chinese postman problem is to find k short-est round trips that covers all links, which corresponds to kvehicles clearing all links. The k-rural Postman Problem isto find k shortest round trips that covers certain links, whichcorresponds to k vehicles clearing a subset of the links.

An important question is the objective function. We couldminimize themakespanof thewhole operation, i.e.,minimizethe maximal time of a vehicle. This leads to the min-max k-Chinese postman problem.We could alsominimize total costfor the clearing, i.e., minimize the sum of the times for thevehicles. This leads to the original k-Chinese postman prob-lem. Minimizing total time will also minimize total distancetraveled when the vehicles are not clearing, which in princi-ple minimizes the pollution caused by the vehicles. The bestobjective function is, we believe, a weighted combination ofthose above.

Our snow removal planning problem therefore turns outto be a new version of the k-Chinese postman problem.More precisely, it is a weighted combination of the origi-nal k-Chinese postman problem, where the total distance is

minimized, and the min-max k-Chinese postman problem,where the maximal distance traveled by a vehicle is mini-mized. The two objectives are here simply added togetherwith certain weights. In practical terms, it is a compromisebetween minimizing the cost for all travel (and equipment)and minimizing the time until all snow is removed.

If there are links that we do not need to treat, we havethe k-rural postman problem. There could be links where thesnow already has been removed, or where the responsibilityof snow removal lies elsewhere.

The computational tests in this paper have been done withthe two tools, Snowplan and Vineopt (implemented by our-selves). In Fig. 1, we show a preliminary run with Snowplan,and in Fig. 2, we show how the network and allocation arebrought intoVineopt.More details about these tools are givenlater.

Finally, we wish to point out that the techniques used inthis paper can be applied to many other multi-vehicle arcrouting applications, not only snow removal.

2 Survey

The Chinese postman problem is a well-known problemwhere each link in a graph needs to be covered. It can besolved optimally and polynomially by a well-knownmethod,[9,10]. The rural postman problem is a similar problem,where, however, only a subset of the links need to be covered.This problem is NP-hard, but there exist efficient heuristics,[5,6,11,13,14].

The k-Chinese postmanproblem is treated in the followingpublications. In [24], complexity results for different ver-sions of the Chinese postman problem are generalized toversions of k-Chinese postman problems. In [1,2], heuristicsand metaheuristics are presented for the min-max k-Chinesepostmanproblem. In [23], newvariants of the k-Chinese post-man problem are presented, together with heuristics for thek-Chinese postman problem. In [3] the min-max k-vehicleswindy rural postman is treated. General references for arcrouting are [7,8].

Winter road maintenance is treated in the followingpapers. In [4], the snow disposal assignment problem ismod-eled as a multiresource generalized assignment problem, anda two-phase heuristic is developed. The paper [21] presentsmodels and methods for partitioning a city into sectors forsnow disposal operations, and for assigning the sectors todisposal sites. The heuristics include a penalty-based assign-ment phase, a two-opt exchange and a districting algorithmof savings-type. The thesis [31] proposes a real-time schedul-ing model for winter road maintenance operations as a largeMIP-model, which is solved by heuristics based on iterativesolving a simplified problem with standard software. Thiswork is continued in [12].

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Fig. 1 Snowplan: Åtvidaberg, G100, q = 4

Rural snow removal is discussed in [20,30], where thefocus lies on finding a set of periodic routing paths during acontinued snowfall. Heuristics, including simulated anneal-ing, are used.

A comprehensive survey of problems related to winterroadmaintenance is found inpapers [26], dealingwith systemdesign for spreading and plowing, [27], dealing with systemdesign for snow disposal, [28], dealing with vehicle rout-ing and depot location for spreading, and [29], dealing withvehicle routing for snow plowing and disposal. Many opti-mization models and solution methodologies are presentedand categorized. The paper [25] presents amodel for the rout-ing of vehicles for snow plowing operations in urban areas.The model is large, and two heuristic approaches are used. Inthe first, several routes are constructed in parallel by sequen-tially solving a multiple vehicle rural postman problem with

vehicle road segment dependencies, turn restrictions, andload balancing. The second uses the approach of cluster-first route-second, and first determines a partition of the arcsinto clusters, with similar work load, and then for each clus-ter solves a hierarchical rural postman problem with classupgrading possibilities, vehicle road segment dependencies,and turn restrictions. This paper is very interesting for us, butthe work is aimed at solving the problem once each winterseason. We wish to be able to solve our problem each day (ifnecessary).

3 Notation

An example of a model for the min–max k-rural postmanproblem is given in [3]. Here we do not plan to use MIP-

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Fig. 2 1045: Vadstena, Vineopt, q = 6

codes for solving the problem, so we will not go into all thedetails of such a model. However, some notation will help inthe description of the heuristic methods.

We will use the following indices: k ∈ Q: the set of vehi-cles, j ∈ L: the set of links (streets), and i ∈ N : the set ofnodes (points, crossings). We will use q = |Q|, n = |N |and m = |L|. The links are assumed to be undirected, i.e.,the combined operation discussed previously can be done ineither direction. Link j has length l j , which is obtained fromthe digital maps.

The set of links that shall be cleared of snow are denotedLC ⊆ L . Often LC = L , i.e., all links in the network need tobe cleared, but sometimes there are links that will be clearedby another operator, or by other types of vehicles, but stillare available to use for transport of our vehicles. As will beclear later, for our method it will not matter much if LC = Lor not.

The value of q, the number of vehicles, may be fixed, butwe also consider the case when it can be varied. We will notmake a model where this parameter is a variable, but willrather do a parametric analysis, i.e., solve the problem fordifferent values of q and compare the results, as there areonly a few interesting values.

We also need the following data:

t j : the time needed for a vehicle to clear link j .d j : the time needed for driving a vehicle along link j

(while not clearing).We will consider the average speed to be vA m/s, so it will

take d j = l j/vA seconds to drive the whole street j . Thetime to clean a road is estimated in average to be p times thelength of the road divided by the speed. A value of p = 2 isreasonable for an ordinary street. This time includes two orthree sweeps (middle, right side, left side) as well as cleaning

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of turning space and/or crossroads. All of this is seen as oneoperation that starts in one end of the street and finishes in theother end. For narrow streets or bicycle paths, p = 1 wouldbe more appropriate.

It then takes t j = pd j = pl j/vA seconds to clean roadj . We will assume that the cost for driving a vehicle isproportional to the time it is driven, so letting cD be thecost for driving one second, the cost for cleaning link j isc j = cDt j = cD pl j/vA.

The total time needed for clearing is t T OT = ∑j∈LC t j ,

which is a constant. The optimization will only concern therest of the solution, which is traveling around with the vehi-cles without removing snow.

We will use f as the fixed cost for using a vehicle, whenconsidering the number of vehicles.

We also need the following:

zR : the time when all vehicles are ready.zUk : the total time vehicle k is used.zCk : the total time vehicle k is used for clearing snow.zLk : the total time vehicle k is used for transportation.

Some relevant objective functions are the following:Minimize final end time: min zR

(the min-max k-Chinese/rural postman problem)Minimize sum of total times for vehicles: min

∑k z

Uk

(the original k-Chinese/rural postman problem)Minimize sum of transport times: min

∑k z

Lk

As there is a fixed proportion between time and cost (cD),the last two objective functions above could just as well beregarded asminimizing the cost. They are often accompaniedby a constraint on zR (the snow must be removed within agiven time frame).

In this paper, we will use an objective function that is aweighted sum of the maximal time for a vehicle and the totalcost for the whole operation.

min wAzR + wC

(

cD∑

k

zUk + f q

)

,

where wA and wC are the weights reflecting the importanceof time relative to cost. IfwA is large, it is important to finishearly, i.e., tominimize the total time until all links are cleared,while ifwC is large, it is more important tominimize the totalcosts for the whole operation.

When we solve the problem, f q will be constant, but wemay solve the problem for a couple of different values of q,and compare the results, and then this termwill be important.

4 Lower bounds

Wewill later describe heuristics forfinding feasible solutions.This yields upper bounds on the optimal objective functionvalue. It can then be useful to obtain some lower bounds on

the optimal objective function value, by solving a relaxationof the problem.

We can then compare the upper and lower bounds to esti-mate the possible potential for improvement. If we are verylucky, the lower bound may be close enough to the upperbound, so that we may conclude that the solution is closeenough to the optimum.

Apossible use for a relaxation is to construct a branch-and-bound method around it. By branching, the lower bound willget closer to the upper bound, and finally indicate optimality.A possible binary branching is to fix that vehicle k should dotask j in one branch and not in the other. Another possibilityis to create q branches for a certain link j , where vehicle kdoes task j in branch k.

The simplest lower bound, denoted by l0, is obtained bydividing the total work that needs to be done by the numbervehicles. Here all transportation is omitted and the need forforming connected cycles is ignored.

A slightly better lower bound is to find the cost of oneChinese postman tour, zP , i.e., the cost for one vehicle todo all, and divide by the number of vehicles. This yieldsthe bound l1 = wAt1/q + wC (cDt1 + f q), where t1 =zE/p + t T OT , since the extra work, zE = zP − t T OT , isonly transportation.

The time for obtaining l0 is negligible, while the timeneeded to obtain l1 is somewhat longer, but still much shorterthan what is needed for solving the problem, i.e., finding agood feasible solution for q > 1. In our framework, wechoose to calculate l1.

5 Solutionmethods

We will focus on heuristic solution methods, such as con-structive heuristics, local search and metaheuristics. Themain parts are the following. First, we make an initial allo-cation, i.e., divide the links among the vehicles. Then wefind a feasible solution, i.e., a route for each vehicle. Afterthis, we try to improve the routes by local search, and bymetaheuristics.

A high-level algorithm for the method is given as Algo-rithm 1. Details will be specified afterwards.

The total cost, which is the sum of the costs for each vehi-cle’s tour plus the fixed costs for vehicle, is computed ateach iteration. Moreover, the time for the solution is com-puted, which is the maximal over the vehicles’ times. Theobjective function value is then computed as a weighted sumof the cost and the time.

Simulated annealing is used for moving to a new alloca-tion, i.e., better allocations are always accepted and worseallocations are accepted with a certain probability.

Steps 3, 9 and 10 can be done in variousways, as describedlater. We also have to choose which of the optional steps

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Algorithm 1 High-level algorithm1: Read the network.2: Construct a starting order for the vehicles.3: Construct an initial allocation of links to vehicles.4: Solve the rural postman problem for each vehicle.5: Optional: Make a vehicle clean all uncleaned links passed by it, even

if they were allocated to another vehicle.6: Evaluate the objective function.7: Optional: Improve the solution by moving cycles from one tour to

another one.8: Optional: Change the allocation to the current solution.9: Change the order for the vehicles.10: Change the allocation of links to vehicles.11: Repeat steps 4–10.12: Save the allocation of links to vehicles.

5, 7 and 8 to do. These choices are governed by param-eters that will be specified below. (Some possibilities thatin preliminary tests in [16] were found to be inferior areremoved.)

Two important vectors are o, which is the order in whichthe vehicles are considered, and κ , which states which vehi-cle should clean each link. More precisely oi is the vehicle(index) treated as the i-th one in the loop, and κ j is the vehicle(index) that is assigned to link j .

5.1 Starting allocation

It can be very important which starting allocation is used.One possible quick use of the method is to use only the start-ing allocation, without subsequent tries to improve it. It iscontrolled by the parameter SV.

0: Start from the current allocation.1: Create a random allocation.2: Read a link set from file and use as a starting allocation.3: Sort in latitude for starting node.4: Sort by distance from the origin to starting node.5: Use k-means, with boundary links allocated to the highest

number.6: Use k-means, with boundary links allocated to the lowest

number.7: Use k-means, with boundary links randomly allocated.8: Use k-means, with boundary links allocated to minimize

distance to center.

Using SV = 0, we can repeatedly try to improve thecurrent solution,while resetting the simulated annealing tem-perature between the runs. The case SV = 2 enables startingfrom a previously constructed allocation. This can be usediteratively to continuewith a previous allocation that has beenimproved by other means (in Vineopt). For SV = 3–4, thelinks are sorted based on position, and the allocation is madefor one vehicle at a time, picking the closest links in order. It

can be seen as a kind of (crude) geographical clustering. ForSV = 5–8, the nodes are clustered in q groups, see Sect. 5.2,thereby inducing an allocation of the links for the vehicles.The difference is how boundary links, i.e., links with the endnodes in different clusters, are treated.

5.2 k-means clustering

Given the coordinates of the nodes N , the k-means clusteringproblem means finding k centers and allocating each node toa center, so that the sum of distances from each node to itsallocated center is minimal. The problem is NP-hard, butefficient heuristics exist, see for example [22].

This can be used for our problem with k = q, i.e., tocreate one cluster for each vehicle. This gives a partitioningof the nodes to the vehicles. Since we are more interestedin the links, we must do the following. If both the startingand ending nodes of a link belongs to the same cluster, thelink is allocated to that cluster. However, if the starting nodeand the ending node of a link belongs to different clusters,we must decide to which cluster of the two the link shouldbelong. In our computational test, we have tried some simpleheuristic ways, namely to allocate the link to the cluster thathas the highest index of these two, or to the lowest index, orrandomly to one of the two clusters.

We have also tried the following. If link j has its twoend nodes in different clusters, s j ∈ A and e j ∈ B, wecalculate the “cost” of letting each cluster cover the othernode, as the distance from each cluster center to the otherendpoint of the link. If this distance is large, the vehicle hasto travel far from its center to include the link. Obviously,this is no exact measure at all, but it may give some indi-cation. We then choose the allocation that minimizes thismeasure.

Let d(A, s j ) denote the distance between center of clus-ter A and node s j , and d(B, e j ) the distance between centerof cluster B and node e j . If link j is allocated to clusterA (i.e., node e j is covered by cluster A), we get the dis-tance d(A, s j ) + l j , and if link j is allocated to cluster B(i.e., node s j is covered by cluster B), we get the distanced(B, e j ) + l j . Obviously, if d(A, s j ) < d(B, e j ), then the“cost” is least if we choose the first alternative. Thus, ifd(A, s j ) < d(B, e j ), then link j is allocated to cluster A,and if d(A, s j ) > d(B, e j ), then link j is allocated to clusterB.

In this paper, we will use “vehicle” and “cluster” inter-changeably, both indexed with k.

Using clustering this way will produce allocations wherevehicles rarely go into each other clusters, which mightseem good. However, in Fig. 3, we give two cycles, onedashed, with lengths on the paths. Swapping the two vehi-cles on the shorter paths as shown to the right in the figure,makes the longest cycle shorter, so it improves the solution.

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34

107

34

107

Fig. 3 Changing allocation

Fig. 4 Moving cycles

Thus, we see that it may be beneficial to let vehicles crosspaths.

5.3 Finding a tour for one vehicle

We recall that routing a vehicle on a given set of undirectedlinks is a rural postman problem, and efficient heuristics forthis problem are presented in [14]. The set of required linksfor vehicle k is simply those links that have κ j = k.

A practical aspect (omitted in the problem formulation) isthat it may take a longer time to turn a vehicle around thanto continue around the block. However, we can do a slighteffort to avoid turning as follows.

When solving a rural postman problem, additionalarcs/edges are added to form a graph where an Euler tourexists. The procedure ends by finding an Euler tour in theextended graph. In this stage, the Euler tour is never unique,as it is a collection of subtours that can be taken in any order.All Euler tours are equally good, so we may choose any oneof them. Then it is possible to avoid a U-turn as much as pos-sible, by simply avoiding going back to the recently visitednode if possible.

5.4 Improve solutions

As an example, in Fig. 4, we first give two tours, one long(blue) and one short dashed (red). Both vehicles pass the

middle node, so one may move the dash-dotted cycle (green)in the second graph from the long to the short, without theneed for any vehicle to move to any other node. The totaldistance is unchanged by this, but the longer cycle becomesshorter, and the shorter cycle becomes longer, so themaximaltime is decreased.

We thus try to improve the solution by finding a cycle ina tour, checking if the cycle contains a node in another tour,and then moving the cycle to that tour. This does not changethe total cost for the solution, as precisely the same subtoursare used. However, moving a subtour from one vehicle toanother may decrease the final time. It is controlled by theparameter IV:

0: Do not move cycles.1: Move cycles, try to move a cycle from an expensive tour

to a cheap one.2: As 1, try all combinations.

5.5 Mark links done

When the tour of the first vehicle is found, one usually findsthat the vehicle passes links that are not allocated to be clearedby that vehicle.

Then it is possible to “disobey” the allocation, and let thatvehicle clear all links it passes. These links are then done,and later vehicles need not pass them. It is not certain thatthis is a good idea, since it could mean that the first vehicledoes a lot of work while the following do less and less. (Herethe difference between c and d matters, i.e., how much fastera vehicle drives without clearing. It is probably a better ideaif this difference is small.). It could, however, be worth to try.

Consider the example in Fig. 5. First, we give the allo-cation for two vehicles, one dashed, and then the cycle of

Fig. 5 Changing allocation

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the first vehicle as a thicker line. Then we change the allo-cation of the dashed links that the first vehicle passes to thatvehicle. After this, the dashed cycle only need to cover themiddle link. Assuming length 1 of each link, the first cyclehas length 6, and the second cycle length 2, which sums upto 8. In the last picture, we show how long the second tourwould be if the allocation was not changed. Then the sec-ond cycle would get length 6, so the sum of lengths wouldbe 12.

The method first finds a tour for the first vehicle, whichpasses all links that are allocated for the vehicle. If MD =1, all links passed by the vehicle are marked as done, i.e.,these links do not need to be treated by other vehicles. If wechange the order in which tours for vehicles are decided, adifferent solution could be generated.

Therefore, the vector o can be changed to get differentsolutions. There are several different approaches to (ran-domly) change o, see below.

Step 8 in the algorithm is only interesting if step 5 is used.It decides if the changed allocation should be saved for lateriterations, RS = 1, or if the old allocation (which was notfollowed) should be kept, RS = 0.

5.6 Change order of vehicles

When finding the routes for the vehicles, we solve a ruralpostman problem for each vehicle, and one question is theorder in which the vehicles are handled, as discussed in theprevious section. Therefore, we consider changing the orderof the vehicles. We use the following possibilities, controlledby parameter ChV.

0: No change of o.1: Swap two random elements in the vector o.2: Create a completely new order.3: Swap the most and least used vehicles.

For ChV = 3, we sum up the number of links allocatedto each vehicle, and switch the positions of the vehicle thathas the most links allocated to it and the vehicles that has theleast. If we are using MD = 1, this often means swappingplaces between the first and the last vehicle, and will make abig difference for the following tours.

5.7 Change allocation

We use simulated annealing to try to improve the allocationof vehicles to links. The neighborhood is defined by eithermoving a link from one vehicle to another, or swapping linksbetween two vehicles. The changes can easily be made byexplicitly working with κ j . A move means changing onevalue. A swap can be seen as changing places of two elementsin the vector.

The changes are controlled by the parameter MV.

0: No change.1: Swap two random links.2: Move a link from a vehicle that does most to a vehicle

that does least.

ForMV = 2, the number of links allocated to each vehicleis calculated, and a link is moved from the vehicle that hasthe highest number of links to the vehicle that has the leastnumber of links.

In the algorithm for simulated annealing, we use thefollowing parameters: MI, number of main iterations, ST,starting temperature, II, number of iterations with fixedtemperature, RC, cooling factor. One iteration of the mainalgorithm is given in algorithm 2. This is just to give thestructure of the method, as we cannot go into all details.

6 Computational details

6.1 Implementational details

The implementation of Snowplan is done in Python, usingNumpy and Scipy (providing the code kmeans). The testswere run on an Acer Aspire X3 X3995 3.4GHz, runningLinux, Fedora Core 22. The machine has four CPUs, butonly one was used in the test runs. For visual aid, Tkinter andnetworkx are used. For solving the rural postman problems,we use a heuristic method described in [14], implemented inC.

We also use the graphic network optimization toolVineopt, written by Kaj Holmberg in Tcl/Tk. It has the abil-ity to show networks and allow changes in them, as well assolve various optimization problems in graphs, especially theChinese postman problem (optimally) and the rural postmanproblem (heuristically).

Vineopt can visualize solutions and link sets (in differ-ent colors), over a map background obtained from Open-StreetMap. It also allows manual changes of link sets,representing the allocations.

6.2 Input data

In our computational tests, we consider the average speed tobe vA = 7.2 km/h, which is 2 m/s. The time to clean a road isestimated in average to be twice the length of the road dividedby the speed, i.e., we use p = 2. With these assumptions, theoperation takes exactly one second per meter, i.e., it takes l jseconds to clean street j if it has length l j . We also give onesecond the cost of one, so times will be equal to costs. Weassume that the fixed cost is 200 for each vehicle. Therefore,the total cost for a solution is the sum of the costs for each

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Algorithm 2 One main iteration1: v ← 02: κ ← κ � current solution3: tmax ← 04: changealloc(tour) � change allocation5: changeorder � change order6: ldone ← 0 � links that are done7: for i = 1, q do � loop over vehicles8: v j ← o[i]9: for j = 1,m do � construct req for vehicle10: if κ[ j] = v j and ldone[ j] = 0 then11: req[ j] ← 112: else13: req[ j] ← 014: writerupin(v j ) � write indata file for rup15: runrup � solve rup16: tourv ← readrupout(v j )17: obj ← calctourcost(tourv)18: tour [v j ] ← tourv19: if MD then � mark passed links done20: aredone(tourv,v j )

21: v ← v + obj + f22: if obj > tmax then23: tmax ← obj � new max time24: v ← wCv + wAtmax

25: if v < vbest or (v = vbest and tmax < tbest ) then26: vbest ← v

27: tbest ← tmax

28: if MD then29: κbest ←revfromtour(κ ,tour)30: else31: κbest ← κ

32: besttour ← tour33: if IV > 0 then � improve solution34: t, v,tour ← improvesol(tour)35: else36: t ← tmax

37: if RS then38: κ ← revfromtour(κ ,tour) � update allocation39: if v < v then � if better go there40: v ← v

41: κ ← κ

42: if v < vbest or (v = vbest and t < tbest ) then43: tobj ← (v − wAt)/wC

44: vbest ← v

45: tbest ← t46: if MD then47: κbest ← revfromtour(κ ,tour)48: else49: κbest ← κ

50: besttour ← tour51: else � or maybe save κ anyway?52: δ ← v − v

53: p ← exp(−δ/T )

54: if random ≤ p then55: v ← v

56: κ ← κ

57: else � back58: κ ← κ

59: i i ← i i + 160: if i i > II then � change temperature61: T ← RC ∗ T62: i i ← 1

vehicle plus number of vehicles times 200. The total objectivefunction value is here the sum of the maximal time and thetotal cost, i.e., wA = wC = 1.

7 Computational tests and evaluation

7.1 Test problems

The methods have been tested on several different probleminstances, based on OpenStreetMap-data, obtained from theInternet. See [18] for details of the data extraction procedure.The networks cover smaller cities aroundLinköping, and alsodifferent parts of Linköping.

A certain filter was applied to the OSM-data, leaving onlystreets (highways) with the following label: “motorway”,“trunk”, “primary”, “secondary” and “tertiary”, but leav-ing out “road”, “cycleway”, “living_street”, “pedestrian”,“footway”, “path” and “residential”. These instances are thusbased on real life networks, with real life distances, but withcertain links removed, and nodes with degree two eliminatedby simply adding the two adjacent links into one, summingup the costs. Therefore, the curvature of the link is not visible,but the distance is correct.

We have, for practical reasons, divided the set into threeparts, depending on the size. The first part contains networkswith less than 200 nodes. They are rather small cities or areas,and give rather easy problems. The second part contains net-works with between 200 and 400 nodes. These problems area bit more time consuming to solve. The third part containsthe most difficult problems, networks with more than 400nodes.

In Table 1, we give the number of nodes and links, thetotal distance of all the links (i.e., the total distance to becleared from snow), the average node degree, and the numberof inhabitants for the cities represented in the test set. Forthe examples based on parts of Linköping, no number ofinhabitants can be specified. (The whole of Linköping hasaround 140,000 inhabitants.)

Pictures of some of the networks are given in Figs. 6, 7, 8and 9.

7.2 Preliminary tests

In Fig. 1, we show a preliminary run with Snowplan. In thegraph at the bottom of the picture, the blue line denotes theactual objective function value for each iteration, the greencross denotes the objective function value after the improve-ment procedure, and the dashed red line denotes the bestobjective function value obtained up to that iteration. Thelight blue horizontal line denotes the lower bound.

In the top of the picture, one can see the parameters thatare possible to change, and their current values. At the top

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Table 1 City instances, sorted in three groups

Name Nodes Links Total dist Deg Inhab

atvid-s 68 95 2028 1.40 –

brokind 179 199 2664 1.11 502

colonia 24 36 131 1.50 –

ekangen 149 184 2629 1.23 2037

mantorp-c 122 145 2243 1.19 –

mantorp 185 226 4152 1.22 3671

rimforsa-o 94 109 1474 1.16 –

skanninge-c 122 152 2292 1.25 –

skanninge-n 62 80 974 1.29 –

skanninge 127 158 2322 1.24 3140

studentryd-e 76 104 543 1.37 –

sturefors 183 212 3283 1.16 2229

valla-liu 160 213 1183 1.33 –

vikingstad 178 213 3390 1.20 2096

atvid 220 290 5940 1.32 6859

borensberg 211 257 4994 1.22 2886

kisa 311 359 4541 1.15 3687

liu 291 394 2713 1.35 –

rimforsa 233 262 2837 1.12 2238

ryd1 382 508 2200 1.33 –

studentryd-int 235 316 1494 1.34 –

studentryd 387 519 2920 1.34 –

linghem 836 1129 6624 1.35 518

ljungsbro 812 1024 14, 013 1.26 6620

malmslatt 489 626 6542 1.28 5214

mjolby 419 524 11, 894 1.25 12,245

ryd-m 1107 1495 10, 910 1.35 –

ryd 952 1291 7903 1.36 –

ryd2 987 1337 9174 1.35 –

soderkoping 666 910 10, 998 1.37 6992

vadstena 581 778 7785 1.34 5613

valla 527 707 5583 1.34 –

left, the total time for each vehicle is shown. There one canidentify the vehicles that takes the longest and the shortesttime, and may consider moving links between them. In thisexample, onemay consider moving links from vehicles 1 and2 to vehicles 3 and/or 4.

In the middle green part, the best objective function valueis shown, together with some more information, such asat which iteration the best value was found, and the lowerbound.

In Fig. 2, we show how the network and allocation arebrought into Vineopt. This allows single links to be changedin Vineopt. The resulting allocation can then be saved andused as starting solution in Snowplan.

VINEOPT

Fig. 6 Åtvidaberg (atvid), 224 nodes, 294 links

VINEOPT

Fig. 7 Linghem, 836 nodes, 1129 links

In [16], we present preliminary tests, made to find inter-esting values of the parameters. (More parameters and valuesare tested than presented here.) One should remember thatthe method contains randomness, so all results should beregarded as approximate. In any case, in these tests, the bestsetting seems to be the following: MD 1, RS 1, SV 5, MV 2,IV 1, ChV 3, II 7, ST 1000, RC 0.3. In words: Mark passedstreets done, revise allocation after this, start with k-means,put border links in the highest number set, move a link from avehicle that does most to one that does least, try all combina-tions ofmoving cycles, swap themost and least used vehicles,try to move a cycle from an expensive tour to a cheap one,try once for each pair of vehicles. In simulated annealing,

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VINEOPT

Fig. 8 Malmslätt (malmslatt), 489 nodes, 626 links

VINEOPT

Fig. 9 Vadstena, 581 nodes, 778 links

do 7 interior iterations (with the same temperature), use starttemperature 1000 and cooling factor 0.3. An observation isthat these parameter settings make the method act more likegreedy heuristics than metaheuristics, i.e., one searches forthe best improvement in every step, and does not do randomchanges and leave the optimization to the metaheuristic. Thereason is probably that each iteration is rather expensive, sowe do not have time to do as many iterations as one wouldprefer in a metaheuristic.

We have chosen a number of interesting settings, i.e., vari-ants of the method to try more. They are shown in Table 2.

Table 2 Settings to try

MD RS SV IV ChV II RC

A 1 1 5 1 3 7 0.3

B 1 1 5 1 3 10 0.5

C 1 1 6 1 3 7 0.3

D 1 1 7 1 3 10 0.5

E 1 1 8 1 3 7 0.3

F 1 1 5 2 3 7 0.3

G 1 0 5 1 3 7 0.3

H 1 1 3 1 3 7 0.3

I 0 0 5 1 3 7 0.3

J 1 1 4 1 3 7 0.3

K 1 1 1 0 3 7 0.3

L 1 1 1 0 2 7 0.3

Additionally, MI = 30, MV = 2, ST = 1000 for all

We will also use the notation Ak, Bk, etc., which denotessetting A with k iterations and so on. For the setting A1, B1etc., the method becomes a one-shot heuristic. If the startingsolution is very good, the succeeding changes may not givemuch improvement, and the question is how many iterationsone should spend on trying to improve the solution.

7.3 Tests of one-stepmethods

In [16], we give results for the settings with only one itera-tion, followed by one round of improvements. We use fourvehicles, and find that the times are very reasonable.

We find that the settings K1 and L1 give bad solutions,while good solutions are obtained by several other settings.Some solutions are actually close to the lower bound, butmost are not very close. Setting E1 gives the best solution for8 instances, followed by G1 and C1, both giving the best for5 instances. A conclusion is that one cannot be sure of goodresults using only one iteration.

7.4 Tests with 30 iterations

In several tables in [16] we present detailed results of ourmain tests with the chosen settings, A–L, using 30 iterations.In Table 3, we give the best and the worst of the results.Checking how many times each setting gave the best objec-tive function value for q = 4 and 30 iterations, we get thefollowing: A 8, G 6, B 4, C 4, E 4, F 4, D 2, H 1, and theothers zero. The randomness makes it hard to pick one “best”setting, but here A seems to be a good choice, since it givesthe best solution for eight problems.

Times are obviously longer here than with one iteration.However, it is still possible to use themethod, fromapracticalperspective, as it is much faster to use this tool than to do itby hand.

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Table 3 Results, best and worstobjectives and times, settingsA-L, q = 4

Name Obj Time Lowb

Min Max Min Max l0 l1

atvid-s B 3842 K 4881 K 1.00 J 1.73 3335 3690

brokind D 5701 K 9134 K 3.02 J 5.12 4130 5225

colonia G 989 K 1042 K 0.69 J 0.96 964 981

ekangen G 4933 L 7265 K 2.30 I 3.92 4086 4707

mantorp-c G 4473 L 7093 L 1.67 H 3.68 3604 4174

mantorp A 7353 K 11477 K 3.36 C 5.75 5990 6948

rimforsa-o B 3337 L 4704 K 1.28 F 1.93 2643 3105

skanninge-c A 4375 L 6095 K 1.68 I 3.39 3665 4259

skanninge-n E 2366 L 2840 L 0.88 G 1.38 2017 2248

skanninge F 4450 K 6098 K 1.79 D 3.02 3703 4300

studentryd-e F 1599 L 1732 L 1.03 H 3.08 1479 1559

sturefors D 6418 L 9398 L 3.24 I 5.69 4904 5826

valla-liu G 2657 K 3323 L 2.59 J 4.56 2279 2533

vikingstad B 7234 K 10,646 K 3.21 B 5.69 5038 6341

atvid G 10,443 K 13,963 K 5.40 B 8.92 8225 9657

borensberg A 9451 K 12,660 K 4.34 E 8.54 7042 8785

kisa C 8903 K 13,040 L 15.06 A 18.09 6476 7981

liu F 5150 L 6353 J 13.81 H 19.08 4191 4931

rimforsa B 5724 L 8837 K 6.18 J 14.07 4346 5266

ryd1 C 3993 L 4767 J 23.80 A 30.48 3550 3926

studentryd-int C 2995 K 3739 K 7.15 D 11.16 2668 2926

studentryd A 5242 L 6307 I 24.93 C 33.61 4450 5074

linghem H 11,132 L 14804 I 171.83 K 274.90 9080 10562

ljungsbro A 23,290 K 32700 I 154.99 K 229.68 18316 22207

malmslatt E 10,411 L 14,130 I 41.85 K 50.67 8978 10187

mjolby A 21,041 L 28,322 I 30.22 B 38.65 15668 18851

ryd-m E 16,868 K 20,340 G 403.35 K 787.02 14437 16159

ryd A 12,137 K 14,486 I 262.40 K 425.07 10678 11910

ryd2 E 13,948 K 16,957 G 286.28 K 522.63 12268 13707

soderkoping A 17,566 K 22,537 G 95.05 K 131.28 14548 16913

vadstena F 12,343 K 15,545 I 61.92 K 91.95 10532 12147

valla G 9517 K 11,486 I 49.57 K 65.81 7779 9039

InTable 4,we investigate the settingA30more thoroughly,by giving the results together with lower bounds and relativeerrors. The quality of the lower bounds may vary betweeninstances.

7.5 Different number of iterations

In Table 5, we give the results for setting A with 1, 5, 30 and100 iterations. The question here is the value of continuediterations. Since the methods contain randomness, we seecases where more iterations give a worse solution. That willobviously not happen within the same run.

Times grow almost linearly with the number of iterations.As there can be several iterationswithout improvement of the

objective function value, improvement of the objective func-tion value is not very regular. Usually, doing 100 iterationsgives better solutions than doing 30 or less, so it is mainly aquestion of how much time one wants to spend.

7.6 Tests withmore vehicles

Wehave also done testwithmore vehicles. In [16],we presenttests with methods A30–L30 and q = 5. The number of bestobjective function values are the following: E 10, G 6, C 4, D3, F 3, A 2, B 1, I 1, and the others zero. The worst objectivefunction values were given by settings K and L. The minimaltimes, however, were given by K and L, and in a few casesG and I. The solution times for the first group is between 1

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Table 4 Results with setting A30, q = 4

Name Obj Time l1 Err

atvid-s 3828.73 1.49 3690.50 0.036

brokind 5766.56 4.24 5225.82 0.094

colonia 1001.79 0.75 981.29 0.020

ekangen 5502.08 3.41 4707.38 0.144

mantorp-c 4612.99 2.36 4174.39 0.095

mantorp 7592.03 4.86 6948.47 0.085

rimforsa-o 3381.52 1.75 3105.75 0.082

skanninge-c 4623.21 2.49 4259.22 0.079

skanninge-n 2324.27 1.31 2248.61 0.033

skanninge 4738.71 2.60 4300.15 0.093

studentryd-e 1602.75 1.45 1559.96 0.027

sturefors 6443.14 4.74 5826.71 0.096

valla-liu 2764.75 4.01 2533.03 0.084

vikingstad 8445.48 4.72 6341.68 0.249

atvid 10, 753.67 8.41 9657.40 0.102

borensberg 10, 337.00 6.30 8785.55 0.150

kisa 8765.30 16.63 7981.47 0.089

liu 5335.31 15.18 4931.24 0.076

rimforsa 5893.20 8.07 5266.41 0.106

ryd1 4040.36 29.37 3926.69 0.028

studentryd-int 3062.97 8.91 2926.23 0.045

studentryd 5298.02 28.83 5074.35 0.042

linghem 11, 800.73 175.34 10, 562.68 0.105

ljungsbro 23, 566.40 159.42 22, 207.49 0.058

malmslatt 10, 450.04 44.52 10, 187.53 0.025

mjolby 21, 320.07 31.40 18, 851.91 0.116

ryd-m 17, 133.13 439.34 16, 159.83 0.057

ryd 12, 610.77 270.47 11, 910.95 0.055

ryd2 14, 768.43 295.82 13, 707.77 0.072

soderkoping 18, 538.84 99.72 16, 913.25 0.088

vadstena 12, 389.38 74.34 12, 147.58 0.020

valla 9529.93 58.79 9039.05 0.052

Err: Relative error to lower bound, (Obj-l1)/Obj

and 10 s, for the second group between 7 and 47 s and for thethird group between 33 and 772 s. Picking a winner is noteasy, but E and G could be candidates.

In Table 6, we present the tests with methods A30–J30 forsix vehicles for the larger networks, as we believe that onewould not use that many vehicles for smaller areas. Pickinga winner here is even harder, but method I is quite fast, whilethe best results were found by C 3, A 2 and G 2.

7.7 Different numbers of vehicles

We have solved a couple of problems for different numbersof vehicles, from one up to eight, and then added a fixed cost

Table 5 Results, objectives, with setting A with 1, 5, 30 and 100 itera-tions

Name A1 A5 A30 A100

atvid-s 4292 4005 4014 3971

brokind 5761 5804 5916 5585

colonia 1021 994 1001 990

ekangen 5077 5542 5607 5334

mantorp-c 4511 4718 4557 4382

mantorp 7598 7458 7353 7544

rimforsa-o 3485 3387 3412 3340

skanninge-c 4492 4638 4375 4616

skanninge-n 2433 2383 2373 2414

skanninge 4712 4651 4470 4671

studentryd-e 1690 1651 1608 1603

sturefors 6477 6495 6988 6428

valla-liu 2718 2676 2692 2624

vikingstad 7180 7561 7574 7287

atvid 10381 10506 10444 10179

borensberg 11423 10062 9451 10065

kisa 9133 9393 9146 9496

liu 5161 5242 5236 5197

rimforsa 6298 5908 5875 5700

ryd1 4110 4008 4094 4082

studentryd-int 3083 2993 3102 3062

studentryd 5439 5412 5242 5275

linghem 11731 11226 11955 –

ljungsbro 23866 23525 23290 –

malmslatt 10867 10492 10658 –

mjolby 21177 21270 21041 –

ryd-m 17561 16912 16994 –

ryd 12612 12782 12137 –

ryd2 14359 14186 14876 –

soderkoping 18017 17935 17566 –

vadstena 12548 12438 12416 –

valla 11094 9648 9897 –

for each vehicle. The results are reported in Table 7 for equalweights on time and cost, and 8 for weight on time twice theweight on cost, i.e., when it is more important to remove thesnow quickly. We used settings A30.

The time for clearing obviously decreases when the num-ber of vehicles increase, but the vehicle cost increases. Onecan then find a minimum of the total cost.

For equal weights on time and cost, we find that for Man-torp the total cost is minimized for four vehicles, while alsosix is good. For Åtvidaberg, four, five or eight vehicles seemto be best. For Vadstena, which is a somewhat larger city, thecost is minimal for eight vehicles. Again we recall that themethods use randomness, so one should not put too much

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Table 6 Results, best and worst objectives and times, settings A-J,q = 6

Name Obj Time Lowb

Min Max Min Max l1

linghem 11, 057 11, 705 299.70 317.15 10311

ljungsbro 22, 485 24, 238 273.34 303.09 1180

malmslatt 10, 296 11, 350 77.31 92.05 9961

mjolby 21, 290 23, 731 57.80 74.90 18048

ryd-m 16, 113 17, 061 667.29 744.05 15535

ryd 11, 935 12, 557 440.63 493.46 11570

ryd2 13, 697 14, 681 488.72 516.21 13247

soderkoping 16, 820 18, 603 170.62 202.87 16239

vadstena 12, 107 12, 534 128.81 140.17 11791

valla 9367 10, 084 98.98 109.23 8889

Table 7 Different numbers of vehicles, with wA = 1 and wC = 1

Veh mantorp atvid vadstena

Obj Time Obj Time Obj Time

1 10037 0.11 14372 0.20 18356 2.72

2 8161 2.22 11412 3.49 14203 43.08

3 7541 3.23 11026 4.98 12956 55.85

4 7233 4.77 10303 7.47 12475 74.48

5 7660 7.31 10404 13.42 12237 95.96

6 7364 11.61 10798 16.34 12306 139.74

7 7730 15.63 10567 25.07 12248 164.33

8 7812 20.14 10327 28.74 12130 211.49

Table 8 Different numbers of vehicles with wA = 2 and wC = 1

Veh mantorp atvid vadstena

Obj Time Obj Time Obj Time

1 14956 0.13 21458 0.31 27434 2.89

2 10513 3.74 14939 5.86 18792 44.77

3 9311 4.85 13137 7.57 16150 58.07

4 9104 7.59 12680 11.07 14837 74.25

5 8735 10.80 12825 14.85 14649 99.06

6 8375 12.65 11787 20.48 13859 130.47

7 8841 17.02 11931 26.31 13886 159.89

8 8874 19.02 12232 31.96 13763 201.42

trust in small differences. Perhaps the best conclusions ofthese tests are that one should not use less than three vehi-cles in Mantorp and not less then four in Åtvidaberg andVadstena.

For a larger weight on time, we find that the minimumfor Mantorp is found at six vehicles, for Åtvidaberg at sixvehicles and for Vadstena at eight vehicles.

We see that the weights on cost versus time are important.Increasing the weight on time (i.e., that the snow is removedquickly) will increase the number of vehicles needed for theminimal total cost.Obviously, using this tool for a certain city,the authorities/users should put some effort in estimating thebest values of these weights.

Estimating the fixed costs for vehicles may be rather dif-ficult, since we are comparing investments for many yearswith daily cost of operations. We will look more at this ques-tion in the future. For now, the fixed costs are not much morethan guesses, but we note that they give reasonable numbersof vehicles in our tests. The results in this paper are just anillustration of how our methods can be used to dimension thefleet for a certain area.

8 Conclusions and future work

The weighted k-rural postman problem appears when plan-ning urban snow removal. We have developed heuristics forthe problem, using metaheuristics, k-means clustering andlocal search improvements by moving cycles. The methodsgive fairly good solutions to real life problems. Solution timesare reasonable and allows real life usage of the code. Thegraphical software gives possibilities for manual improve-ments.

By adding a fixed cost for each vehicle, we can comparethe costs when using different numbers of vehicles.

Future research includes improved local search by mov-ing cycles between non-adjacent tours, moving paths, andimproved metaheuristics. We will also use the results of thisproblem for a more detailed routing for one vehicle at a time.

Open Access This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate creditto the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.

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